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Bioleaching kinetics of a spent refinery

catalyst using Aspergillus niger at optimal conditions

F. Amiri, S. M. Mousavi, S. Yaghmaei and M. Barati

Version Post-print/Accepted Manuscript

Citation (published version)

Amiri, F., S. M. Mousavi, S. Yaghmaei, and M. Barati. "Bioleaching kinetics of a spent refinery catalyst using Aspergillus niger at optimal conditions." Biochemical engineering journal 67 (2012): 208-217. DOI: 10.1016/j.bej.2012.06.011

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1

Bioleaching kinetics of a spent refinery catalyst using 1

Aspergillus niger at optimal condition 2

F. Amiri a,b, S.M. Mousavi c,*, S. Yaghmaei a,** , M. Barati b 3

a Department of Chemical and Petroleum Engineering, Center of Excellence, Development and 4

Strategic Plants for Bioprocess Technology, Sharif University of Technology, Tehran, Iran 5

b Department of Material Science and Engineering, University of Toronto, Toronto, Canada 6

c Biotechnology Group, Chemical Engineering Department, Tarbiat Modares University, Tehran, 7

Iran 8

9

Abstract 10

The kinetics of bioleaching of Mo, Ni, and Al from spent hydrocracking catalyst, using 11

Aspergillus niger was studied. The four most effective bioleaching variables were selected 12

in accordance with the Plackett–Burman design and were further optimized via central 13

composite design (CCD). The optimal values of the variables for maximum multi–metal 14

bioleaching were as follows: particle size (+150-212 µm), sucrose (93.8 g/L), pulp density 15

(3 %w/v), and pH (7). The maximum metals recoveries corresponding to these conditions 16

were (99.5 ± 0.4) % Mo, (45.8 ± 1.2) % Ni, and (13.9 ± 0.1) % Al. The relatively low Ni 17

extraction was attributed to the precipitation of Ni in the presence of oxalic acid. Under the 18

* Corresponding author: Biotechnology Group, Chemical Engineering Department, Tarbiat Modares

University, Tehran, Iran. Tel: +9821 82884917, Fax: +9821 82884931, E-mail: mousavi_m@modares.ac.ir

** Corresponding author: Tel. : +9821 66166430, E -mail : yaghmaei@sharif.edu

2

optimal conditions, the fungus growth was found to be higher in the presence of spent 1

catalyst than that in the catalyst–free medium. Determinations of the organic acid 2

concentration showed noticeable variation during bioleaching, particularly for gluconic acid. 3

Accordingly, a modified form of shrinking core model was used to take these variations into 4

account. The predictions by the model showed good consistency with the experimental 5

results, suggesting that diffusion of bioleaching agent through the solid matrix was the rate–6

controlling step. 7

Key words: Bioleaching; Waste Treatment; Extraction Kinetics; Spent catalyst; 8

Optimization; Filamentous Fungi 9

10

1. Introduction 11

Ceaseless exploitation of resources in the world has led to the exhaustion of high grade ore 12

[1]. Heavy metals such as Mo, Ni, W, and V are strategic metals widely used in the 13

production of special grades of steel; they are also used as catalysts in petroleum, 14

petrochemical and chemical industries [2]. The gradual depletion of ores containing heavy 15

metals, coupled with an increasing demand have promoted the search for secondary 16

resources such as waste materials and byproducts, including spent catalysts [3]. In the 17

petroleum refining operations, solid catalysts are extensively used to improve the process 18

efficiency [1, 2]. Spent hydrotreating and hydrorefining catalysts have been identified as 19

hazardous wastes by USEPA since 1999 [3]. Hydrocracking catalysts, mainly made of 20

molybdenum (Mo) or tungsten (W) with nickel (Ni) promoter supported on a porous Al2O3/ 21

SiO2 base are commonly used in petroleum refining processes for producing light oils from 22

3

the cracking of heavy crude oil. However, catalysts may be deactivated by contamination of 1

the active surface with elements that are deposited during the cracking reactions (e.g., S, C, 2

V, Fe, Ni, Si, Zn, and As). Deactivation of catalysts also occurs due to structural change by 3

thermal degradation, phase separation, or phase transformation that restricts their 4

reactivation. In such cases, the spent catalyst is replaced with fresh catalyst [3, 4]. As a result, 5

large amounts of deactivated hydrocracking catalysts (i.e. spent catalyst) are generated that 6

warrants their treatment before land disposal in order to meet the environmental regulations. 7

This is primarily required because the predominant elements such as V, Ni, Mo, and Co, in 8

the catalysts are toxic and can be easily leached out with water, leading to generation of a 9

secondary pollution [3-6]. 10

Worldwide, several companies are involved in metal reclamation from spent 11

hydroprocessing catalysts; their technologies are based on two main approaches: 12

hydrometallurgy or pyrometallurgy. Through the hydrometallurgical approach, the metals 13

are leached out by means of catalysis with an acid or a base, while pyrometallurgical 14

processing requires high temperature treatment of the materials through processes such as 15

roasting and smelting [2, 7]. Due to several drawbacks of the conventional techniques such 16

as high energy costs and generation of environmental pollutants, bioleaching processes have 17

been developed as an alternative method [8, 9, 10]. 18

Bioleaching is based on the ability of microorganisms to transform solid compounds into 19

soluble elements which can subsequently be recovered. In comparison with conventional 20

technologies, bioleaching is associated with advantages such as simple operation, reduced 21

operating cost, and smaller environmental footprint [11-15]. Most leaching active fungi 22

4

which have been isolated and used for this purpose belong to the genera Aspergillus or 1

Penicillium with the ability to excrete large amounts of leaching agent such as organic acids 2

[13-16]. 3

The nature and the amount of organic acid excreted by fungi are mainly influenced by: 1- 4

the pH of medium, 2- the buffering capacity of the medium, 3- the carbon source, 4- the 5

presence or absence of certain heavy metals and trace elements, 5- balances of nitrogen and 6

phosphate, 6- the temperature of the medium, 7- pre-culture period and inoculum used 8- 7

resistance of microorganisms to metal ions, 9- physical and chemical states of the solid 8

residue, 10- liquid to solid ratio, and 11- bioleaching period [10, 17]. With the objective of 9

maximizing the metal recovery in an industrial operation, process optimization is critical in 10

such a system with numerous influential factors. To achieve this with the minimum number 11

of experimental or plant trials, it is thus necessary to analyze the process with an initial 12

screening design prior to optimization [18]. 13

The statistical screening method offers several advantages over the conventional approaches 14

including: 1- being rapid and reliable, 2- identifying the effective factors, 3- taking into 15

account the interactions among the factors, and 4- reducing the total number of experiments 16

[10, 19]. Plackett–Burman (PB) design is an effective technique which screens the 17

components that significantly influence the process and eliminates the insignificant 18

components in order to obtain a smaller, more manageable set of factors. It has been applied 19

to a number of processes including pre–treatment of macroalgae for Pb, Cr and Al 20

determination by graphite furnace atomic absorption spectrometry (GF-AAS) [20], 21

5

polysaccharide and ergosterol production from Agaricus brasiliensis by fermentation 1

process [21] and bioethanol production [22]. 2

Response surface methodology (RSM), one of the global optimization methods, is a 3

collection of statistical and mathematical techniques useful for development, improvement, 4

and optimization of multivariable processes. It also has important applications in designs, 5

development and formulation of new products, as well as improvement of existing product 6

designs [23, 24]. RSM has been successfully employed to optimize the compositions of 7

microbiological media for biosorption of heavy metal [25], improving fermentation process 8

[23] and waste treatment [24]. Although the method is widely used for other processes, there 9

are only few examples in the literature involving RSM for bioleaching of solid waste [10]. 10

In kinetics studies of bioleaching systems, generally, it is assumed that the concentration of 11

the leaching agent is constant [9, 11]. However, the concentration of the biometabolites in 12

the bioleaching systems varies due to the presence of microorganism, hence such assumption 13

may lead to inaccurate simulation of the bioleaching process using conventional kinetic 14

models. This assumption was modified by Haghshenas et al. [26] and the variation of ferrous 15

iron ion during sphalerite bioleaching by Acidithiobacillus ferrooxidans was considered in 16

the shrinking core model. 17

In our previous study [27], a two–step bioleaching process was investigated in which the 18

fungus was added first and the spent catalyst was introduced after initiation of biometabolite 19

production (as opposed to simultaneous addition of both). It showed that increasing the pulp 20

density does not give rise to a lower metal yield, which may be advantageously used to 21

6

operate under higher rates of catalyst addition. However, the metal recoveries were generally 1

low, warranting an optimization study to increase the recoveries. 2

The present study was undertaken to investigate the kinetics of two-step bioleaching of the 3

metals from the spent catalyst under optimal condition. We report for the first time a 4

sequential optimization strategy for biological metal extraction from spent catalyst by 5

Aspergillus niger through statistically designed experiments. First, the effective parameters 6

were screened using PB two-level factorial design and then optimization of the significant 7

parameters was carried out using RSM to maximize metal extraction from spent catalyst by 8

Aspergillus niger efficiently. Afterward, a kinetics study was performed on bioleaching of 9

the spent catalyst under optimal conditions and the profile of metal recovery percent, organic 10

acid production together with pH, and fungal dry weight were discussed. Finally, the fungal 11

leaching rate controlling step was determined for the first time using a modified shrinking 12

core model (SCM) that takes into account the changes in the bioleaching agent concentration 13

with time. 14

15

2. Materials and methods 16

2.1. Micro-organism 17

The micro-organism used in this study is Aspergillus niger BBRC-20018. This micro-18

organism was provided by the Biochemical and Bioenvironmental Research Center (BBRC), 19

Sharif University of Technology (Iran). 20

21

2.2. Spent catalyst 22

7

Spent hydrocracking catalyst (Albemarle KF-1015-LH Mo/Ni/Al2O3/SiO2) was provided by 1

National Iranian Oil Refining & Distribution Company (NIORDC) and this material was 2

used to prepare all the samples. The as–received spent catalyst with a black covering was 3

first pre-treated by heating in a furnace at 600 °C for 4 h. The decoked spent catalyst was 4

gently grinded and sieved to separate the fraction with desired particle size. Chemical 5

digestion [16] indicated that the spent hydrocracking catalyst consisted of 6.4% Mo, 2.4% 6

Ni, and 24% Al. 7

8

2.3. Bioleaching experiment according to the experimental design 9

Prior to bioleaching experiments, the fungal strain was acclimatized to metal ions and spent 10

catalyst in a prolonged adaptation period in a procedure described elsewhere [16]. To obtain 11

sufficient numbers of spores, the adapted fungus Aspergillus niger was cultured in a PDA 12

(potato dextrose agar, 3.9% (w/v)) slant and incubated at 30 °C for 5 days. The mature 13

conidia were then washed from the surface of the PDA medium using sterilized saline 14

solution (9 g/L NaCl). The spores were counted using a Neubauer counting chamber and 15

adjusted using sterilized saline to approximately 107 spores/mL. Specified quantity of the 16

spore suspension by experiment design, was added to a 500 mL Erlenmeyer flask containing 17

100 mL of sucrose medium. The concentrations of sucrose, NaNO3, KH2PO4 and yeast 18

extract were determined by the experimental design. The trace compounds MgSO4·7H2O 19

and KCl were used at a constant concentration of 0.025 g/L. Prior to inoculation, pH of the 20

growth medium was adjusted according to experimental design, and the medium was 21

autoclaved at 121 °C for 15 min. The flasks were agitated in an incubator with specified 22

8

orbital shaking and temperature. After the perscribed preculture time, the spent catalyst of a 1

specific particle size was added to the medium, and the incubation was continued to various 2

time intervals up to 30 days. 3

4

2.4. Analytical methods 5

After the desired bioleaching time, the culture from each flask was filtered and the filtrate 6

was analyzed for the concentration of organic acids and various metal ions. Organic acids 7

(i.e. citric, oxalic and gluconic acids) were analyzed using high performance liquid 8

chromatography (HPLC) HP 1100 series, with variable wavelength detector (VWD) at 210 9

nm. The concentrations of metal ions were measured using an ICP-AES (Varian, model: 10

LIBERTY–RL). Multi elements standards (Merck) were used for calibration. In order to 11

extract and measure the biomass-accumulated and associated metals, the method described 12

by Santhiya and Ting (2006) [15] was used. The loss of liquid because of evaporation was 13

estimated from control (cell free) experiments and was included in the calculations. The 14

residue (i.e., the fungal biomass with the bioleached catalyst) obtained from the filter paper 15

was carefully transferred to a pre-weighed evaporating dish and was dried at 80 °C for 24 h. 16

The dried residue was ashed at 500 °C for 4 h to determine the dry weight of the biomass. 17

The pH of the leached liquor was measured using a digital pH meter (Metrohm, Model: 827 18

pH lab). 19

A scanning electron microscope was used to observe the morphology of the catalysts. The 20

spent and bioleached catalysts were used for the FESEM (Field Emission SEM) 21

examination. The spent and bioleached catalysts were mounted with silver paste on 22

9

aluminum stubs, then coated with 300–400 Å Au in a sputtering unit and finally examined 1

in the FESEM. Image acquisition was conducted under high vacuum and at an accelerating 2

voltage of 15 kV. 3

4

2.5. Statistical approaches 5

Plackett–Burman (PB) designs are very useful for selecting the most important factors from 6

a long list of candidate factors such as carbon source (sucrose), inorganic salts (NaNO3, 7

KH2PO4), nitrogen source (yeast extract), cultivation parameters (preculture time, inoculum 8

size, temperature, pH, shaking rate) and catalyst parameters (particle size, pulp density). This 9

methodology assumes that the important main effects will be much larger than two-factor 10

interactions. PB designs require fewer runs than a comparable fractional design. Under the 11

conditions when the number of runs is higher than the number of variables (at this stage of 12

analysis, there are 11 variables and 12 trials) better resolution is achieved than a saturated 13

design. Consequently, this technique can be used to identify the more important independent 14

variables for the optimization step [16]. Because Mo was the most important heavy metal 15

present in the spent catalyst, the influence of 11 variables on its bioleaching was investigated 16

using the methodology of PB. Each independent variable is tested at two levels, a high (+) 17

and a low (–) level, as shown in Table 1. The critical ranges of selected parameters were 18

determined through preliminary experiments based on the literature review [13, 17, 25]. The 19

ranges should be neither short nor wide to represent the effect of factor’s value changing 20

properly. 21

10

In the present study, central composite design (CCD) was used as the most widely used 1

response surface method in the experimental design and optimization. The approach is well 2

suited for fitting a function using the least-squares method and usually works well for the 3

process optimization [28, 29]. CCD was applied using Design-Expert® 7.0.0. 4

Generally, the CCD consists of a 2k factorial runs with 2k axial runs and x0 number of center 5

points, where k is the number of independent variables. The center points were used to 6

determine the experimental error and the reproducibility of the data. The independent 7

variables are coded where the low and high levels are represented as -2 and +2, respectively. 8

The axial points are located at (± α, 0, 0), (0, ±α, 0) and (0, 0, ±α) where α is the distance of 9

the axial point from center, which is given by α = 2 n/4 (for 4 factors n = 4, α = 2). The 10

experimental sequence was randomized in order to minimize the effects of the uncontrolled 11

factors. 12

In the optimization process, the responses can be simply related to the screened factors by 13

linear or quadratic models. A quadratic model, which also includes the linear model, is given 14

as (Eq. (1)): 15

𝑌 = 𝛽0 +∑ 𝛽𝑖𝑘𝑖=1 𝑋𝑖 + ∑ ∑ 𝛽𝑖𝑗

𝑘𝑗=1

𝑘𝑖=1 𝑋𝑖𝑋𝑗 (1) 16

where Y is the predicted response, k is the number of variables, β0 is the model constant, βi 17

is the linear coefficient, βii is the quadratic coefficient, βij is the interaction coefficient. 18

In this study, the experimental plan consisted of 30 trials and the value of the dependent 19

response was the mean of two replications. The relationships and interrelationships of the 20

variables were determined by fitting a model to the dataset obtained from 30 experiments. 21

The mathematical model generated during RSM implementation was validated by 22

11

conducting experiment on given optimal medium setting. Statistical analysis of the data was 1

performed by analysis of variance (ANOVA). 2

2.7. Kinetics investigation of bioleaching at optimal condition 3

The variations with time of of metal recovery, organic acid production, fungus dry weight 4

and pH were measured under the optimal conditions determined earlier. By assuming a 5

spherical form for the spent catalyst, and acidolysis as the main mechanism involving in 6

fungal leaching, the rate controlling step of metal extraction was determined by the 7

shrinking core model (SCM). 8

9

3. Results and discussion 10

3.1. Screening of bioleaching significant variables 11

A total of eleven variables were analyzed with regard to their effects on metal recovery 12

percent using a PB design (Table 1). The design matrix selected for the screening of 13

significant variables for biological metal extraction and the corresponding response are 14

shown in Table 2. The adequacy of the model was calculated, and the variables showing 15

statistically significant effects were screened via their ANOVA p-values. Factors yielding 16

P-values of less than 0.05 were considered to have significant effects on the response, and 17

were therefore selected for further optimization studies. Particle size, with a probability 18

value of 0.015, was determined to be the most significant factor, followed by sucrose (0.022), 19

pulp density (0.026), and pH (0.046). All other insignificant variables were neglected, and 20

the optimum levels of the four variables were further determined by CCD design. 21

22

12

3.2. Optimization of the screened variables 1

The significant variables utilized were as follows: particle size, pulp density, sucrose and 2

pH, each of which was assessed at five coded levels (-2, -1, 0, +1, and +2), as is shown in 3

Table 3. To determine the optimum levels of these variables, the response surface 4

methodology (RSM), using a central composite design (CCD), was adopted. A 24 fractional 5

factorial CCD for four independent variables each at five levels, with eight axis points and 6

six replicates at the centre points leading to 30 experiments was used in this study. The full 7

experimental plan with regard to their values in actual and coded form and the corresponding 8

results of CCD experiments are provided in Table 4. The response values (Y) in each trial 9

were the average of the duplicates. 10

Statistical results of the ANOVA for response surface quadratic models are presented in 11

Table 5. 12

According to the statistical data for the quadratic models presented in Table 5, the Model F-13

values of 22.1, 20.7 and 20.2 implied that the models were significant and there was only a 14

0.01% chance that a ‘Model F-value’ could occur because of noise. The ‘Prob > F’ values 15

for the models which were less than 0.05 (<0.0001) indicated that the models were 16

statistically significant with a confidence interval of 99.99%. The “lack of fit tests” compares 17

the residual error to the “Pure Error” from replicated experimental design points. The p-18

values, greater than 0.05, for models indicate that lack of fit for the model was insignificant. 19

In designed experiments, R2 is always increasing with the addition of variables to the model 20

whether the additional variable is statistically significant or not. Thus it is possible to obtain 21

models having large R2 values that yield poor predictions of new observations or estimates 22

13

of the mean response. Using an adjusted R2 is preferred as presented in Table 5. In general, 1

the adjusted R2 is not always increasing as variables are added to the model. In fact, if 2

unnecessary terms are added, the value of R2adj often decreases [30]. According to Table 5, 3

the values of R2 are evaluated as 0.954, 0.951 and 0.950 while that of R2 adj were 0.911, 4

0.905 and 0.903 for the models Y1A, Y2A and Y3A, respectively. The R2 and R2adj coefficients 5

in this study ensured a satisfactory adjustment of the quadratic model to the experimental 6

data. The coefficient of variance (CV) for the Mo, Ni and Al recovery percent has been found 7

to be (%): 3.36, 3.81, and 5.21. The CV as the ratio of the standard deviation to the mean-8

value of the observed response (as a percentage) is a measure of reproducibility of the model 9

and as a general rule a model can be considered reasonably reproducible if its CV is not 10

greater than 10%. The models showed no lack of fit and the adequate precision value which 11

gives a measure of the ‘‘signal-to-noise ratio’’ was found to be in the range of 14–20, which 12

indicate an adequate signal. A ratio greater than 4 is desirable [30]. 13

According to numerical optimization by Design-Expert 7.0.0, several optimum conditions 14

for different objectives are presented in Table 6. 15

16

3.2.1. Mo recovery model 17

Data from the 30-batch run was analyzed using the Design-Expert 7.0.0 software, and the 18

following second-order quadratic model for Mo recovery (%) was obtained (Eq. (2)): 19

Y1A = 65.21 + 0.19A + 0.21B - 6.44C - 1.29D - 3.06AB + 0.64AC - 1.96AD -1.56BC + 20

2.28BD - 1.04CD + 1.14A2 + 0.054B2 - 1.24C2 + 1.06D2 (2) 21

14

The significance of each coefficient was determined by p-values through statistical analysis. 1

The smaller the p-value indicates the higher significance of the corresponding coefficient 2

[31]. Model term with p- value <0.05 including: the independent variables of C (pulp 3

density) and D (pH), the interaction variables of AB (particle size × sucrose), AD (particle 4

size × pH), BC (sucrose × pulp density), and BD (sucrose × pH) as well as quadratic variables 5

of A2, C2 and D2 are considered as significant variables. On the other hand, the highest 6

absolute coefficients of terms C, AB and C2, respectively among the independent, interaction 7

and quadratic terms shows their higher importance on Mo recovery. 8

Fig. 1a shows the normal probability plot for the empirical model for Mo recovery. The data 9

fall on a straight line in the plot of the residuals, which represents a normal distribution and 10

thus supports the adequacy of the least-squares fit. The residual plot (Fig. 1b) which shows 11

equal scatter of the residual data above and below the x-axis indicates that the variance was 12

independent of the value of the Mo recovery, again supporting the adequacy of the least-13

squares fit. 14

According to Table 6, there are two optimal conditions for maximum Mo recovery (%). one 15

is: particle size of 169µm (+150-212 µm), sucrose concentration of 90.6 g/L, pulp density 16

of 4.72 %w/v and pH of 7.22 and the other is: particle size of 19.6 µm (-38 µm), sucrose 17

concentration of 167 g/L, pulp density of 3.35 %w/v and pH of 10.7, with both conditions 18

yielding Mo recovery of 100%. 19

Fig. 2 shows the second-order response surface plot for the Mo recovery (%) as a function 20

of particle size and sucrose concentration as the most important interaction when pulp 21

density and pH were at fixed optimal values of 4.72 %w/v and 7.22 (Fig. 2a) and 3.35 %w/v 22

15

and 10.7 (Fig. 2b), respectively. There was a nearly linear relationship between Mo recovery 1

and sucrose concentration, while the particle size had a nonlinear effect on the Mo 2

bioleaching. As presented in the Fig. 2a, the highest Mo recovery corresponded to the lowest 3

concentration of sucrose (90 g/L) and the largest particle size (+150-212 µm). The binding 4

of leaching material to the fungal mycelium has been mentioned as a “common problem” in 5

fungal leaching process by Burgstaller and Schinner [17]. Decreasing the particle size might 6

result in occupying more available binding sites [32] and bioaccumulation in A. niger. 7

Therefore, bioaccumulation of spent catalyst by fungus might have a potential inhibition 8

effect on the metals concentration in the actual leaching liquor. A more detailed investigation 9

will be carried out in the future. 10

As it is evident in Fig. 2a, the increased sucrose concentration at the highest particle size 11

(+150-212 µm) results in decreasing metal recovery. It is assumed that the increased organic 12

acids production due to the increasing in the sucrose concentration have an inhibition effect 13

on fungal growth [34]. On the other hand, according to Fig. 2b the greatest Mo dissolution 14

occurs at the highest concentration of sucrose (170 g/L) and the smallest particle size (-38 15

µm). The decreased particle size results in increasing the bioleaching rate due to the 16

increased surface area [9] before bioaccumulation; therefore the biometabolites involved in 17

bioleaching and their inhibiting effect on the fungal growth is diminished. 18

In view of the economic benefits, low sucrose concentration and large particle size is more 19

favourable as optimal condition. 20

21

3.2.2. Ni and Al recovery models 22

16

The experimental Ni and Mo recovery from the spent catalyst were fitted to a second-order 1

quadratic equation, as follows. 2

Y2A = 36.65 - 0.20A +0.68B - 4.25C -1.24D - 1.45AB - 0.63AC - 0.57AD - 0.27BC + 0.12BD 3

-0.31CD -0.42A2 - 0.51B2 + 0.84C2 + 0.29D2 (3) 4

Y3A =11.40 - 0.097A + 0.34B - 1.59C - 0.28D - 0.46AB - 0.64AC - 0.15AD + 0.013BC + 5

0.12BD -0.33CD -0.35A2 -0.31B2 -0.32C2 -0.14D2 (4) 6

The ANOVA analysis of the optimization study indicated that the common important model 7

terms including independent variables B, C, and D, interaction variable AB, and cubic 8

variable C2 were significant in the two models (P < 0.05). The effect of pulp density (P < 9

0.0001) was determined to be more significant than the effects of the other variables. The 10

interactions between particle size and sucrose were significant, as was shown by the low P-11

value (<0.005) for the interactive terms. 12

A negative sign for the coefficients of variable C in the fitted models (Eq. (3), (4) and (5)) 13

indicated that the level of metal recovery increased with decreasing levels of factor C. 14

Increasing the pulp density results in low dissolved oxygen and a high concentration of the 15

heavy metals which inhibit the fungal growth and organic acid production [10]. 16

The normal probability plot of the residuals as well as the residual versus the fitted values 17

plot for the empirical Ni and Al recovery model (not shown) presented similar profiles as 18

that in the Mo recovery (Fig. 1a and b). The results again support the assumption of a normal 19

distribution in these models. 20

17

The optimal conditions for maximum Ni and Al concentration (shown in Table 6), 1

respectively, are particle size of 174 and 175 µm, sucrose concentration of 105 and 110 g/l, 2

pulp density of 1 and 1%w/v, and pH of 7 and 8.86. 3

The optimum metal recovery values (Table 6; case 2, 3) are in agreement with the results 4

obtained from the two-dimensional contour plots at the optimal conditions of Ni and Al 5

bioleaching (Fig. 3). 6

7

3.3. Kinetics investigation of metal dissolution at optimal condition 8

According to Table 6, Cases 4 and 5 describe the conditions under which optimal multi metal 9

recovery can be achieved. For these two cases, the metal recovery efficiencies for Ni and Al 10

are less compared to the previous scenarios, however, the lower sucrose concentration and 11

larger pulp densities are beneficial from economic and productivity aspects. In that regard, 12

Case 5 conditions with minimized sucrose concentration and maximized pulp density 13

present a more attractive alternative to Case 4, hence its conditions were considered as the 14

overall optimal conditions for investigation of multi metal bioleaching kinetics. The 15

achieved optimal conditions were in contrast with that of previous studies with maximized 16

sucrose concentration and minimized pulp density and particle size which were not 17

economically attractive [9, 10]. 18

19

3.3.1. Metal recovery and organic acid production over time at optimal condition 20

Fig. 4 shows the extent of leaching of different metals under optimal conditions against the 21

leaching time. As it is obvious in this figure, the leaching kinetics followed a two–domain 22

18

behavior, an initial rapid period that turns into a slow regime after about 3–7 days. Similar 1

behaviour was also observed in a few previous reports on bacterial leaching of spent catalyst 2

[11, 33]. The trends were attributed to different reasons such as depletion of easily available 3

reacting species, formation of a product layer or a diffusion limited like intra-particles, or a 4

combination of these phenomena [33]. The initial faster rate was observed within 7, 3, and 5

3d of initiation of bioleaching for Mo, Ni, and Al, respectively, resulting in 88.0% Mo, 6

40.3% Ni, and 13.8% Al recoveries. In the following slow bioleaching regime, only 7

additional 11.5% Mo, 5.5% Ni, and 5.2% Al were recovered in 21, 23, and 23 days 8

respectively. The ultimate metal recoveries were as follows: (99.5±0.4) % Mo, (45.8±1.2) 9

% Ni, and (13.9±0.1) % Al, which conformed well to the models presented above. The 10

achieved Mo extraction yield was the highest value compared with previous studies [5, 6, 8, 11

9, 11-16, 27, 34]. A significantly higher Mo recovery (%) over Ni was expected because the 12

optimization was performed with the significant variables affected on Mo bioleaching. 13

Nevertheless, the obtained Ni recovery (%) under these conditions is considerable compared 14

to some reported results [13]. Control (cell free) experiments under optimal conditions were 15

also performed which accounted for leaching of (34.0 ± 1.2) % Mo, (8.80 ± 1.0) % Ni, and 16

(0.51 ± 0.1) % Al. A comparison of these control results with those in our previous studies 17

[16, 34] shows that the initially higher pH of the control medium has accounted for the higher 18

Mo, Ni, and Al extraction. 19

The organic acid concentration produced by Aspergillus niger under the optimal conditions 20

over time is shown in Fig. 5. The gluconic acid was the main produced organic acid and its 21

concentration reached 367 mM in the 7th day of bioleaching. The oxalic and citric acid 22

19

maximum concentrations also occurred at the same time after which a decreasing trend was 1

observed in the organic acid concentration. This may be associated to the consumption of 2

the acid in the bioleaching process as well as a drop in the organic acid production, caused 3

by the decrease in the fungus growth rate. It was noticed that the increase in the leaching of 4

the metals (mainly Mo) paralleled the increase in the concentration of organic acids (mainly 5

gluconic acid). This phenomenon indicated that the biogenically produced organic acids 6

played a direct and important role in the bioleaching process, and confirmed that depletion 7

of bioleaching agents could be one reason in decreasing the rate of metal extraction. A 8

similar finding has been reported by Aung and Ting (2005) [13]. 9

Lower Ni recovery compared to the previous investigations [11, 12, 14-16, 27, 34] is 10

possibly because of the production of oxalic acid which precipitates a fraction of the leached 11

nickel as nickel oxalate with a reported low solubility. Clearly, the optimal condition for Ni 12

extraction requires generation of citric acid rather than oxalic acid [27]. 13

Al constitutes the base of catalyst and its extraction is very challenging, especially when it 14

exists as ∝ −𝐴𝑙2𝑂3. This phase is completely inert towards acids because it is originally 15

produced by calcinations of Al(OH)3 above 1000 °C which gives it great stability towards 16

acids [35]. However, the extraction of Al from spent catalyst is not that critical since the 17

residue can be reused for production of the catalyst base. 18

19

3.3.2. Fungus dry weight and pH changes over time at optimal condition 20

The fungal dry weigh changes (under optimal conditions and in the catalyst-free medium) 21

as well as pH changes (under optimal condition and in cell-free control) are shown in Fig. 6. 22

20

As it is evident from this figure, the amount of fungal growth and its rate under optimal 1

conditions was higher than that in the absence of catalyst. It implies that the inhibitory effect 2

of produced metabolites such as organic acid is reduced in the presence of spent catalyst by 3

involving these metabolites in bioleaching. Consequently, well adapted fungus grew more 4

and faster in the presence of spent catalyst under the optimal conditions. This was also 5

observed in our previous study by Penicillium simplicissimum [34]. 6

The maximum biomass concentrations at different pulp densities (2-12 %w/v) were in the 7

range of 15.0-39.2 g/L with the highest and the least values belonging to optimum and 12% 8

w/v pulp density, respectively. At 13% pulp density, the fungus ceased to grow. The 9

decreased biomass concentration at higher pulp densities compared to the optimum value (3 10

%w/v) is attributed to the strengthened inhibitory effect of the toxic metal from the spent 11

catalyst and the restricted oxygen mass transfer; as also noted by Chen and Lin [36]. 12

Fig. 6 also shows the changes in pH during bioleaching by A. niger under optimal conditions 13

and in cell-free control. In the bioleaching process, the pH decreased to 3.46 after two days 14

which is due to the initiation of organic acid production (Fig. 5), and the spent catalyst was 15

then added to the medium. The introduction of the spent catalyst to the medium caused an 16

increase in the pH value to 3.61. On the other hand, introducing the spent catalyst into the 17

cell–free control caused the pH to drop from 7.0 to 4.8. The lower level of pH in the 18

bioleaching medium compared to that in the cell–free control is because of the organic acid 19

production by the fungus [37]. The pH changes were not significant over time in the cell–20

free control which is because there were no considerable changes in the control medium 21

composition (no producing agent). The observed increasing trend in the pH of the 22

21

bioleaching medium may be due to the organic acid consumption during the bioleaching 1

(Fig. 5). The negligible amount of citric acid production is because during the bioleaching 2

process the pH of the medium remained above 3 while the optimum pH for citric acid 3

production is 2-3 [38]. 4

5

3.3.3. Determination of the rate controlling step 6

In heterogeneous solid–fluid reactions, at least, the following three sequential steps occur 7

[39]: (a) diffusion of reactants through the fluid film surrounding the solid; (b) diffusion of 8

reactants through the solid shell, and (c) chemical reaction. The reactions pertaining to this 9

study are acidolysis and complexolysis as the two main reactions responsible in mobilization 10

of metals. Since the bioleaching was carried out under vigorous shaking condition, it may 11

be assumed that the rate of the overall reaction is not limited by the mass transfer of the 12

lixiviant through the fluid film, i.e. step (a). 13

In acidolysis, the most important fungal leaching mechanism under acidic conditions, the 14

oxygen atoms covering the surface of a metal compound are protonated readily. The protons 15

and the oxygen combine with water and the metal is therefore detached from the surface. 16

The term complexolysis or ligand induced metal solubilization means that a metal ion is 17

solubilized due to the complexing capacity of a molecule. Complexolysis plays two roles: 18

1- enhancing the solubility of a metal ion which has been solubilized via acidolysis [17], and 19

2- facilitating the detachment of metals species from the surface through ligand exchange 20

that polarizes the critical bonds. This latter mechanism is slower than solubilization with 21

protons and become dominant under weaker acidic conditions [17, 40, 41]. Based on the 22

22

acidic pH of the bioleaching medium (Fig. 6) extraction of metals from the solid matrix is 1

considered to be mainly related to the production of protons. 2

Organic acids dissociates only slightly in pure water. Reactions of the dissociation of organic 3

acids were considered in equilibrium form to indicate that the products are not heavily 4

favored. That is mainly because of the instability of the anion which recombines with the 5

hydrogen ion. However, involving the products of organic acid dissociation in the 6

bioleaching (e.g. protons in acidolysis and anions in complexolysis) may push the 7

equilibrium equation toward the right side which results in nearly complete organic acid 8

dissociation. With this assumption, dissociation of one mole of gluconic acid, oxalic acid, 9

and citric acid produces 1, 2, and 3 moles of protons, respectively. 10

The kinetics of the extraction of Mo (the most important heavy metal present in the spent 11

catalyst) was further investigated. The reaction involved in the dissolution of MoO3 by 12

proton attack is presented below. 13

𝑀𝑜𝑂3 + 6𝐻+ → 𝑀𝑜6+ + 3𝐻2𝑂 (5) 14

Due to the fungal activity, the concentration of the organic acids varies during a bioleaching 15

process; therefore, to take into account the variation of protons during bioleaching process, 16

the shrinking core models for diffusion (Eq. (6)) and chemical reaction (Eq. (7)) control 17

should be used as follows [39]: 18

𝐸(𝑋𝐵) = 1 − 2 3⁄ 𝑋𝐵(𝑡) − (1 − 𝑋𝐵(𝑡))2 3⁄

= 2𝑏𝐷𝑒 𝜌𝐵𝑅2(1 − 𝜀)⁄ ∫ 𝐶𝐴

𝑡

0𝑑𝑡 (6) 19

𝐺(𝑋𝐵) = 1 − (1 − 𝑋𝐵(𝑡))1 3⁄ = 𝑏𝑘" 𝜌𝐵𝑅(1 − 𝜀)⁄ ∫ 𝐶𝐴𝑑𝑡

𝑡

0 (7) 20

where XB is the fractional conversion (Mo recovery), 𝜌𝐵 is the molar density of MoO3 in the 21

solid, CA is the proton concentration in the solution ( theoretically estimated from the organic 22

23

acid concentration, M), De is the diffusion coefficient of the proton, R is the radius of the 1

particle, and ε is the porosity of the spent catalyst particle. ∫ 𝐶𝐴𝑡

0𝑑𝑡, is considered to be a 2

function of time, F(t). The values of F(t) at any time were calculated from the area underneath 3

the graph of proton concentration, that in turn was estimated from the variations of organic 4

acids concentration with time. 5

The lines representing the left sides of Equations (6) and (7), as well as the corresponding 6

experimental values are plotted against ∫ 𝐶𝐴𝑡

0𝑑𝑡 and t in Fig. 7 a and b respectively. Time 7

zero corresponds to the time for addition of spent catalyst. The figures show that the kinetic 8

model in which the variation of proton with time is taken into account (Fig. 7a) predicts the 9

experimental data much better than the model in which variation of proton with time is 10

assumed negligible; this was also reported by Haghshenas et al. [26] for variation of ferrous 11

iron ion during sphalerite bacterial leaching by Acidithiobacillus ferrooxidans (Fig. 7b). 12

A comparison between the measurements and the values predicted by the modified 13

correlations (Fig. 7a) suggests that the diffusion model (R2=0.91) fits better with the results 14

than the chemical reaction model (R2=0.77). This is an indication that the overall dissolution 15

kinetics is likely controlled by the diffusion step. This may be attributed to either the low 16

porosity of the de-coked spent catalyst, which is due to the metal deposition and structural 17

changes during the refining process, or the relatively higher rate of acidolysis reaction. 18

19

20

3.3.4. Surface morphology variation of the spent catalyst before and after bioleaching 21

24

FESEM photomicrographs of the catalyst samples before and after the leaching are displayed 1

in Fig. 8. The bioleached sample was calcined at 500 °C for 4 hours prior to the FESEM 2

examination. The precipitates on the spent catalyst surface, seen in Fig. 8a, are probably 3

metal deposits formed during the refining process, and a cause for deactivation of the 4

catalyst. A similar structure was observed in our previous study [16]. The small patches 5

seen on the surface of the bioleached spent catalyst (Fig. 8b) could be attributed to nickel 6

oxide which resulted from insoluble nickel oxalate by burning in the furnace and it turned 7

the color of the bioleached spent catalyst residue into light green. 8

9

4. Conclusions 10

The effective parameters of bioleaching of spent hydrocracking catalyst by Apergillus niger 11

were screened prior to optimization to reach maximum multi metal extraction. Screening 12

and optimization were carried out using Plackett–Burman and Central Composite Design 13

approaches. Four variables (particle size, sucrose concentration, pulp density, and pH) were 14

identified as signficant through screening and were selected for further optimization of Mo, 15

Ni and Al recovery. The optimal particle size for highest multi- metal recovery was found 16

to be the largest (+150-212 µm). This unexpected finding was attributed to increased chance 17

of bioaccumulation of smaller size of spent catalyst in the fungal biomass before efficient 18

bioleaching. The optimal value for sucrose concentration (93.8 g/L) was 1.1 fold smaller 19

than that of the Bosshard’s medium [42]. The achieved optimal conditions in this study, with 20

the largest particle size and the nearly lowest sucrose concentration were favourable in view 21

of the lesser process cost. The obtained Mo recovery was comparable to the chemical 22

25

leaching results and was the highest in comparison with that of previous bioleaching studies. 1

The optimum conditions for maximum Mo recovery was accompanied with low Ni yields, 2

because of the considerable amounts of oxalic acid generation, which in turn gives rise to 3

precipitation of Ni as nickel oxalate. The reduction in the rate of metal extraction after an 4

initially fast period was found to be related to the depletion of organic acids. The dissolution 5

rate of Mo under optimal conditions was higher than that of Ni, and Al. With the assumption 6

of acidolysis as the main mechanism in the fungal leaching under acidic condition, and 7

complete dissociation of organic acid in the bioleaching medium, Mo fungal leaching 8

kinetics was investigated using a modified shrinking core model that incorporates the 9

variations of bioleaching agent (i.e. proton) with time into the correlation. It was found that 10

the diffusion of bioleaching agent into the solid matrix is the rate-controlling step. That was 11

related to either low porosity of the spent catalyst or the relatively higher rate of acidolysis 12

reaction than that of diffusion. 13

14

Acknowledgments 15

The financial support of this project (contract No. 88-1098) by National Iranian Oil Refining 16

& Distribution Company (NIORDC) is appreciated. Special thanks are due to Ms. Zahra 17

Ghabadinejad for her kind cooperation in the microbial work. The authors are grateful to 18

Stat-Ease, Minneapolis, MN, USA, for the provision of the Design Expert package. 19

20

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12

13

14

15

16

17

18

19

20

Table and Figure captions 21

32

Table 1: Experimental variables at different levels used for bioleaching of spent catalyst by 1

Aspergillus niger using Plackett–Burman design 2

Table 2: Twelve-trial Plackett–Burman design matrix for eleven variables with coded values 3

together with the observed Mo recovery (two replicates were used) 4

Table 3: Quantitative value of the coded parameter levels 5

Table 4: Parameter levels of central composite design (coded value) 6

Table 5: Statistical results of the ANOVA 7

Table 6: Optimum process conditions for different objectives according to numerical 8

optimization by Design- Expert 7.0.0 9

10

Fig. 1: (a) normal probability plot, and (b) residual plot for Mo bioleaching model. 11

Fig. 2: Surface plot of Mo recovery percent versus particle size and sucrose concentration, 12

pH and pulp density were kept at fixed value of a) 7.22 and 4.72 %w/v, and b) 10.7 and 3.35 13

%w/v, respectively. 14

Fig. 3: Contour plot of (a) Ni and (b) Al recovery percent versus particle size and sucrose 15

concentration, pH and pulp density were kept at optimal value according to case 2 and 3 16

(Table 6), respectively. 17

Fig. 4: Metal bioleaching yield versus time under optimal conditions according to case 5 18

(Table 6) 19

Fig. 5: Organic acid concentration versus time under optimal conditions according to case 5 20

(Table 6) 21

Fig. 6: Variations of fungal dry weight and pH with time 22

33

Fig.7: Comparison between measurements and correlations expressing diffusion and 1

chemical reaction controlled regimes for bioleaching of Mo: (a) the variation of proton with 2

time is taken into account (b) variation of proton with time is assumed negligible. 3

Fig 8: FESEM photomicrograph of spent and bioleached catalysts: (a) spent catalyst 4

(30000× magnification), (b) bioleached catalyst (30000× magnification) 5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

34

1

2

3

4

Table 1 5

S. no. Variables Units Symbol

code

Experimental values

Lower Higher

1 Temperature °C A 22 35

2 pH - B 5 9

3 Shaking rate rpm C 110 160

4 Particle size µm D +38-75 +150-212

5 Pulp density % w/v E 1 7

6 Inoculation size % v/v F 2 10

7 Preculture time day G 2 10

8 Sucrose g/L H 80 120

9 NaNO3 g/L J 1 2

10 KH2PO4 g/L K 0.25 0.75

11 Yeast extract g/L L 0.5 3

6

7

8

9

10

11

12

13

14

15

35

1

2

3

4

5

6

Table 2 7

Run oder Experimental values Metal recovery (%)

A B C D E F G H J K L Mo

1 +1 +1 +1 -1 -1 -1 +1 -1 +1 +1 -1 58.7±2.3

2 +1 +1 -1 -1 -1 +1 -1 +1 +1 -1 +1 73.9±1.5

3 +1 -1 +1 +1 -1 +1 +1 +1 -1 -1 -1 53.8±0.9

4 +1 -1 +1 +1 +1 -1 -1 -1 +1 -1 +1 54.3±1.1

5 -1 +1 -1 +1 +1 -1 +1 +1 +1 -1 -1 65.2±3.0

6 -1 +1 +1 -1 +1 +1 +1 -1 -1 -1 +1 63.7±1.3

7 -1 -1 -1 +1 -1 +1 +1 -1 +1 +1 +1 46.0±2.4

8 +1 -1 -1 -1 +1 -1 +1 +1 -1 +1 +1 67.7±2.1

9 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 50.4±1.5

10 -1 -1 +1 -1 +1 +1 -1 +1 +1 +1 -1 65.5±1.2

11 -1 +1 +1 +1 -1 -1 -1 +1 -1 +1 +1 51.8±2.1

12 +1 +1 -1 +1 +1 +1 -1 -1 -1 +1 -1 60.7±0.8

8

9

10

11

12

13

14

15

36

1

2

3

4

5

6

7

8

9

Table 3 10

Symbol

code

Variables Units Levels

-2 -1 0 +1 +2

A Particle

size

µm 15

(-38)

55

(+38-75)

95

(+75-106)

135

(+106-150)

175

(+150-212)

B Pulp

density

%w/v 1 2.5 4 5.5 7

C Sucrose g/L 90 110 130 150 170

D pH g/L 7 8 9 10 11

11

12

13

14

15

16

17

18

19

20

21

22

37

Table 4 1

Run

order

Coded levels Metal recovery

percent

A B C D Mo Ni Al

1 1 1 1 1 57.6 29.0 7.2

2 -1 -1 -1 -1 65.7 38.8 10.3

3 -1 -1 -1 1 67.6 37.0 10.8

4 0 0 -2 0 72.3 49.7 13.0

5 1 -1 1 1 55.8 31.5 7.7

6 1 -1 1 -1 71.5 35.6 8.5

7 0 -2 0 0 67.5 34.1 9.5

8 1 1 1 -1 58.3 33.0 8.6

9 0 0 0 0 65.3 36.2 12.0

10 0 0 0 0 65.7 36.1 10.9

11 0 0 0 0 65.4 37.5 11.9

12 0 0 0 2 66.4 34.9 9.7

13 -1 1 -1 1 82.3 42.6 12.7

14 1 1 -1 -1 71.1 44.2 12.5

15 1 -1 -1 -1 78.9 41.6 12.8

16 0 0 0 0 65.3 35.2 10.6

17 1 1 -1 1 71.3 39.1 12.5

18 -2 0 0 0 70.5 36.7 10.6

19 0 0 0 0 61.9 37.5 11.9

20 0 0 2 0 47.9 30.5 7.2

21 2 0 0 0 68.8 33.4 9.4

22 -1 -1 1 1 54.6 29.0 7.6

23 -1 -1 1 -1 59.6 32.7 9.2

24 1 -1 -1 1 69.8 41.2 12.6

25 -1 1 1 1 61.7 36.0 10.1

26 0 0 0 -2 72.2 40.8 12.0

27 0 0 0 0 67.7 37.6 11.1

28 -1 1 -1 -1 76.6 41.8 11.5

29 -1 1 1 -1 58.3 35.6 10.0

30 0 2 0 0 63.1 35.4 10.8

2

3

4

38

1

2

3

4

5

6

7

Table 5 8

Statistical result Y1A Y2A Y3A

Model F-value 22.1 20.7 20.2

Model Prob>F <0.0001 <0.0001 <0.0001

Lack of fit F-value 1.62 2.42 0.68

Lack of fit Prob>F 0.311 0.171 0.721

R-Squared 0.954 0.951 0.950

Adj R-Squared 0.911 0.905 0.903

C.V % 3.36 3.81 5.21

Adeq. Precision 22.1 19.9 16.4

9

10

11

12

13

14

15

16

17

18

19

20

21

39

1

2

3

4

Table 6 5

Case Target Particle size

(µm)

Sucrose

(g/L)

Pulp density

(%w/v) pH

Recovery

(%)

1 Mo Maximize 169

19.6

90.6

167

4.72

3.35

7.22

10.7

100

100

2 Ni Maximize 174 105 1.00 7.00 55.1

3 Al Maximize 175 110 1.00 8.86 14.5

4 Mo

Ni

Al

Maximize

Maximize

Maximize

175 100 2.50 7.00

100

48.3

13.3

5 Mo

Ni

Al

Sucrose

Pulp density

Maximize

Maximize

Maximize

Minimize

Maximize

164 93.8 3.00 7.00

100

45.9

12.2

6

7

8

9

(a) 10

40

1

(b) 2

3

Fig. 1 4

(a) 5

41

1

(b) 2

3

Fig. 2 4

(a) 5

42

1

2 (b) 3

4

Fig. 3 5 6

43

1

2

3

4

5

6

7

Fig. 4 8

9

10

11

12

13

14

0

20

40

60

80

100

0 5 10 15 20 25 30

Met

al

extr

act

ion

(%

)

Time (Days)

Mo Al Ni

44

1

2

3

4

5

6

Fig. 5 7

8

9

10

11

12

13

14

0

50

100

150

200

250

300

350

400

0 5 10 15 20 25 30

Org

an

ic a

cid

con

c. (

mM

)

Time (Days)

Oxalic acid Gluconic acid Citric acid

45

1

2

3

4

5

6

7

Fig. 6 8

9

10

11

12

13

3

3.5

4

4.5

5

5.5

6

6.5

7

0

5

10

15

20

25

30

35

40

0 5 10 15 20 25 30

pH

Fu

ngu

s d

ry w

eigh

t (g

/L)

Time (Days)

Fungal dry weight (at optimal condition)

Fungal dry weight (catalyst-free medium)

pH (at optiml condition)

pH (Cell-free control)

46

(a) 1 2

3 4 5

(b) 6

7

Fig. 7 8

G(XB)= 0.099F(t)

R² = 0.77

E(XB)= 0.036F(t)

R² = 0.91

0

0.2

0.4

0.6

0.8

1

0.00 1.50 3.00 4.50 6.00 7.50 9.00

E(X

B)=

1-2

/3X

B(t

)-(1

-XB(t

))2/3

&

G(X

B)=

1-(

1-X

B(t

))1/3

F(t)=∫t0CA dt

Chemical reaction model Diffusion model

E(XB) = 0.033t

R² = 0.66

E(XB) = 0.012t

R² = 0.83

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30

E(X

B)=

1-2

/3X

B(t

)-(1

-XB(t

))2/3

&

G(X

B)=

1-(

1-X

B(t

))1/3

Time (day)

Chemical reaction model Diffusion model

47

(a) 1

2

(b) 3

4

Fig. 8 5